Paradoxically, the rotation curve of the nearest galaxy remains poorly
known. Extinction is too large to observe the stars and too small to
observe the gas. It is preferable to observe the gas, either at
21 cm or at 2.7 mm,
because it extends at much greater radii. Thus we must rely on the
corotation of both the stellar and the gaseous systems, an assumption
that is not always justified, as mentioned previously. The tangent-point
method to
obtain the rotation curve for R < R0, with
R0 being the solar
galactocentric distance, is well known and need not be repeated here in
detail.

The points of the circle with a Galactic Centre-Sun diameter are
characterized by a radial velocity from the Sun equal to their
rotation velocity, and this velocity is determined by the fact that it
corresponds to the largest redshift (in the first quadrant) or the
largest blueshift (in the fourth quadrant). The different values at
the points of this circle give us the rotation curve.

However, this method only provides the rotation curve out to 8
kpc, but to analyze our dark halo and the mass of the Milky Way itself, this
is too small. To extend the rotation curve to larger galactocentric
distances, a variety of objects have been observed. These objects have
to be bright, to be observed from afar, their distance must be accurately
determined (which in practice is the largest source of error) and their
radial velocities must be easily obtainable. Carbon stars, OB stars,
planetary
nebulae, cepheids and HII regions have been used to study the outer
Galaxy, but the errors are large and the maximum distance is less than
2R0. A complete account of these attempts was given in
the review by
Fich and Tremaine (1991)
and will not be repeated here. There
is a crucial date (1965) prior to which, as reviewed by
Schmidt (1965),
it was thought that the outer rotation curve was Keplerian and the
estimated mass of the Milky Way was about
2 × 1011M. After this year, various authors
began to realize
that the outer curve was more or less flat, and the conclusion that
our Milky Way may contain large amounts of dark matter became more
and more widely accepted.

The best method to investigate the outer rotation curve was proposed
by Merrifield (1992),
who considered that a ring in the
Milky Way with constant vertical scale length, hz, has
a variable
angular size when seen from the Sun; or in his own words "much as
a person standing in a volcano might estimate his or her location
within the crater from the variations in the apparent height of the
walls in different directions". It is easily obtained that

(22)

where vr is the radial velocity from the Sun, (R) is the
rotation at a point with galactocentric distance R, is
the rotation velocity at R = R0 and l
and b are the galactic
longitude and latitude. Therefore, if we have a data cube
Tb(vr, l, b), where
Tb is the HI brightness temperature, it is
possible to divide the cube into slices with constant W(R)
as defined in
the above equation. W(R) only depends on R and we may use hz to
know exactly what value of R we are speaking about. From our
location in the Galaxy, the HI layer of thickness hz
at some point
of radius R will present an angular size in galactic latitude of

(23)

If we then take a constant-W slice, obtain the variation in angular
width as a function of longitude l, and compare it by this formula,
we can calculate it by fitting the value of
R/R0 (and even
hz/R0)
of the slice, and hence obtain
vr(R/R0) and
hz(R/R0).

There are some inherent problems. The orbits must be circular and
vertical shear must be absent, i.e. vr should not
depend on z. The
galactic warp introduces further complications, although these can be
overcome. In such a way, Merrifield was able to reach points in the
Milky Way rotation curve out to about 20 kpc, or 2.5
R0, with an unprecedented degree of precision.

The results greatly depend on the values of R0 and
. Merrifield proposed
R0 = 7.90.8kpc and
= 20010kms-1, rather lower than usually
recommended in other works,
to match other kinematic constraints and in line with the rotation
curves of similar galaxies. More recently, Honma and
Sofue (1996,
1997)
have used Merrifield's method to estimate the
rotation curve, the geometry of the halo and the total mass of the
Milky Way, investigating their uncertainties. Errors in R0 are
relatively unimportant because they just change the scaling in the
radial direction, but changes in produce highly different
interpretations of our hypothetical halo. In
Fig. 11 we plot
their results for three different values of : 220, 200 and
180 km s-1.

Figure 11. Overall rotation curves of the
Galaxy for
= 220kms-1 (filled circles),
= 200kms-1 (open
circles), and
= 180kms-1 (open
triangles). The data for
the inner rotation curve were taken from
Fich et al. (1989).
The outer
rotation curve are those obtained by Merrifield's method. The error
bars are indicated only for
= 220kms-1, and are almost
the same for the three cases. From
Honma and Sofue (1997).
Courtesy of the Astronomical Society of Japan.

The rotation velocity decreases beyond 2R0 for all
cases. Reasonable Keplerian fits are obtained for
R2R0 if
is in the range 200-207 kms-1. If
< 200kms-1 the curve
declines faster than Keplerian.

Using
= 220kms-1, as recommended by
the IAU, the outer
rotation curve rises between
R = 1.1R0 and 2R0, which is
uncommon
in other galaxies of the same type, having a flat curve
within the optical disk. To obtain a flat rotation curve,
should be as small as 192 km/s.

R0 and are related to Oort's constants A and B (note
A - B = /R0), which are fairly well
determined.
(
/R0 could also be directly
determined by means of the VLBI
determination of the proper motion of Sgr B2, taking two quasars
behind as reference. Accurate data, in this respect, will be available
in the near future.
Honma and Sofue (1996)
propose
= 200km and
therefore
R0 = 7.6kpc, based on this result and those
obtained by other
authors also claiming lower and R0.

Assuming a spherical mass distribution they obtain for the mass of the
Milky Way a low value of
2.00.3 × 1011M, which
is close to the early estimates.

The Keplerian rotation curve does not require dark matter beyond
2R0, but it would still be necessary within
2R0, because an
exponential disk has a rotation curve declining beyond 2.2r, when r
is the radial scale length, in conflict with the flat rotation out to
2R0. However, the dark matter needed could be much
less than
previously calculated. On the other hand, the shape of the dark halo
would differ greatly from that theoretically deduced.